TSTP Solution File: SEU813^1 by Satallax---3.5
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%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SEU813^1 : TPTP v8.1.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Tue Jul 19 14:08:30 EDT 2022
% Result : Theorem 15.11s 15.26s
% Output : Proof 16.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : SEU813^1 : TPTP v8.1.0. Released v3.7.0.
% 0.13/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.14/0.36 % Computer : n011.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 600
% 0.14/0.36 % DateTime : Sun Jun 19 09:59:08 EDT 2022
% 0.14/0.36 % CPUTime :
% 15.11/15.26 % SZS status Theorem
% 15.11/15.26 % Mode: mode507:USE_SINE=true:SINE_TOLERANCE=3.0:SINE_GENERALITY_THRESHOLD=0:SINE_RANK_LIMIT=1.:SINE_DEPTH=1
% 15.11/15.26 % Inferences: 0
% 15.11/15.26 % SZS output start Proof
% 15.11/15.26 thf(ty_kfst, type, kfst : ($i>$i)).
% 15.11/15.26 thf(ty_binintersect, type, binintersect : ($i>$i>$i)).
% 15.11/15.26 thf(ty_singleton, type, singleton : ($i>$o)).
% 15.11/15.26 thf(ty_if, type, if : ($i>$o>$i>$i>$i)).
% 15.11/15.26 thf(ty_subset, type, subset : ($i>$i>$o)).
% 15.11/15.26 thf(ty_eigen__2, type, eigen__2 : $i).
% 15.11/15.26 thf(ty_setunion, type, setunion : ($i>$i)).
% 15.11/15.26 thf(ty_iskpair, type, iskpair : ($i>$o)).
% 15.11/15.26 thf(ty_surjFuncSet, type, surjFuncSet : ($i>$i>$i)).
% 15.11/15.26 thf(ty_breln1invset, type, breln1invset : ($i>$i>$i)).
% 15.11/15.26 thf(ty_surjective, type, surjective : ($i>$i>$i>$o)).
% 15.11/15.26 thf(ty_eigen__1, type, eigen__1 : $i).
% 15.11/15.26 thf(ty_eigen__0, type, eigen__0 : $i).
% 15.11/15.26 thf(ty_cartprod, type, cartprod : ($i>$i>$i)).
% 15.11/15.26 thf(ty_breln, type, breln : ($i>$i>$i>$o)).
% 15.11/15.26 thf(ty_injective, type, injective : ($i>$i>$i>$o)).
% 15.11/15.26 thf(ty_emptyset, type, emptyset : $i).
% 15.11/15.26 thf(ty_breln1, type, breln1 : ($i>$i>$o)).
% 15.11/15.26 thf(ty_setminus, type, setminus : ($i>$i>$i)).
% 15.11/15.26 thf(ty_kpair, type, kpair : ($i>$i>$i)).
% 15.11/15.26 thf(ty_descr, type, descr : (($i>$o)>$i)).
% 15.11/15.26 thf(ty_reflwellordering, type, reflwellordering : ($i>$i>$o)).
% 15.11/15.26 thf(ty_breln1compset, type, breln1compset : ($i>$i>$i>$i)).
% 15.11/15.26 thf(ty_breln1Set, type, breln1Set : ($i>$i)).
% 15.11/15.26 thf(ty_ap, type, ap : ($i>$i>$i>$i>$i)).
% 15.11/15.26 thf(ty_funcSet, type, funcSet : ($i>$i>$i)).
% 15.11/15.26 thf(ty_symdiff, type, symdiff : ($i>$i>$i)).
% 15.11/15.26 thf(ty_omegaS, type, omegaS : ($i>$i)).
% 15.11/15.26 thf(ty_nonempty, type, nonempty : ($i>$o)).
% 15.11/15.26 thf(ty_dsetconstr, type, dsetconstr : ($i>($i>$o)>$i)).
% 15.11/15.26 thf(ty_injFuncSet, type, injFuncSet : ($i>$i>$i)).
% 15.11/15.26 thf(ty_powerset, type, powerset : ($i>$i)).
% 15.11/15.26 thf(ty_omega, type, omega : $i).
% 15.11/15.26 thf(ty_ex1, type, ex1 : ($i>($i>$o)>$o)).
% 15.11/15.26 thf(ty_ksnd, type, ksnd : ($i>$i)).
% 15.11/15.26 thf(ty_exu, type, exu : (($i>$o)>$o)).
% 15.11/15.26 thf(ty_prop2set, type, prop2set : ($o>$i)).
% 15.11/15.26 thf(ty_set2prop, type, set2prop : ($i>$o)).
% 15.11/15.26 thf(ty_lam, type, lam : ($i>$i>($i>$i)>$i)).
% 15.11/15.26 thf(ty_in, type, in : ($i>$i>$o)).
% 15.11/15.26 thf(ty_func, type, func : ($i>$i>$i>$o)).
% 15.11/15.26 thf(ty_binunion, type, binunion : ($i>$i>$i)).
% 15.11/15.26 thf(ty_dpsetconstr, type, dpsetconstr : ($i>$i>($i>$i>$o)>$i)).
% 15.11/15.26 thf(ty_image1, type, image1 : ($i>($i>$i)>$i)).
% 15.11/15.26 thf(ty_setadjoin, type, setadjoin : ($i>$i>$i)).
% 15.11/15.26 thf(def_setextAx,definition,(setextAx = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))))).
% 15.11/15.26 thf(def_emptysetAx,definition,(emptysetAx = (![X1:$i]:(~(((in @ X1) @ emptyset)))))).
% 15.11/15.26 thf(def_setadjoinAx,definition,(setadjoinAx = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))))).
% 15.11/15.26 thf(def_powersetAx,definition,(powersetAx = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))))).
% 15.11/15.26 thf(def_setunionAx,definition,(setunionAx = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))))).
% 15.11/15.26 thf(def_omega0Ax,definition,(omega0Ax = ((in @ emptyset) @ omega))).
% 15.11/15.26 thf(def_omegaSAx,definition,(omegaSAx = (![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))))).
% 15.11/15.26 thf(def_omegaIndAx,definition,(omegaIndAx = (![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))))).
% 15.11/15.26 thf(def_replAx,definition,(replAx = (![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))))).
% 15.11/15.26 thf(def_foundationAx,definition,(foundationAx = (![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))))).
% 15.11/15.26 thf(def_wellorderingAx,definition,(wellorderingAx = (![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))))).
% 15.11/15.26 thf(def_descrp,definition,(descrp = (![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))))).
% 15.11/15.26 thf(def_dsetconstrI,definition,(dsetconstrI = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_dsetconstrEL,definition,(dsetconstrEL = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))))).
% 15.11/15.26 thf(def_dsetconstrER,definition,(dsetconstrER = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))))).
% 15.11/15.26 thf(def_exuE1,definition,(exuE1 = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))))).
% 15.11/15.26 thf(def_prop2setE,definition,(prop2setE = (![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))))).
% 15.11/15.26 thf(def_emptysetE,definition,(emptysetE = (![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))))).
% 15.11/15.26 thf(def_emptysetimpfalse,definition,(emptysetimpfalse = (![X1:$i]:(~(((in @ X1) @ emptyset)))))).
% 15.11/15.26 thf(def_notinemptyset,definition,(notinemptyset = (![X1:$i]:(~(((in @ X1) @ emptyset)))))).
% 15.11/15.26 thf(def_exuE3e,definition,(exuE3e = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))))).
% 15.11/15.26 thf(def_setext,definition,(setext = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))))).
% 15.11/15.26 thf(def_emptyI,definition,(emptyI = (![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))))).
% 15.11/15.26 thf(def_noeltsimpempty,definition,(noeltsimpempty = (![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))))).
% 15.11/15.26 thf(def_setbeta,definition,(setbeta = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))))).
% 15.11/15.26 thf(def_nonemptyE1,definition,(nonemptyE1 = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))))).
% 15.11/15.26 thf(def_nonemptyI,definition,(nonemptyI = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_nonemptyI1,definition,(nonemptyI1 = (![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))))).
% 15.11/15.26 thf(def_setadjoinIL,definition,(setadjoinIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))))).
% 15.11/15.26 thf(def_emptyinunitempty,definition,(emptyinunitempty = ((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)))).
% 15.11/15.26 thf(def_setadjoinIR,definition,(setadjoinIR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_setadjoinE,definition,(setadjoinE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))))).
% 15.11/15.26 thf(def_setadjoinOr,definition,(setadjoinOr = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))))).
% 15.11/15.26 thf(def_setoftrueEq,definition,(setoftrueEq = (![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)))).
% 15.11/15.26 thf(def_powersetI,definition,(powersetI = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))))).
% 15.11/15.26 thf(def_emptyinPowerset,definition,(emptyinPowerset = (![X1:$i]:((in @ emptyset) @ (powerset @ X1))))).
% 15.11/15.26 thf(def_emptyInPowerset,definition,(emptyInPowerset = (![X1:$i]:((in @ emptyset) @ (powerset @ X1))))).
% 15.11/15.26 thf(def_powersetE,definition,(powersetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))))).
% 15.11/15.26 thf(def_setunionI,definition,(setunionI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))))).
% 15.11/15.26 thf(def_setunionE,definition,(setunionE = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))))).
% 15.11/15.26 thf(def_subPowSU,definition,(subPowSU = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))))).
% 15.11/15.26 thf(def_exuE2,definition,(exuE2 = (![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))))).
% 15.11/15.26 thf(def_nonemptyImpWitness,definition,(nonemptyImpWitness = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))))).
% 15.11/15.26 thf(def_uniqinunit,definition,(uniqinunit = (![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))))).
% 15.11/15.26 thf(def_notinsingleton,definition,(notinsingleton = (![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))))).
% 15.11/15.26 thf(def_eqinunit,definition,(eqinunit = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))))).
% 15.11/15.26 thf(def_singletonsswitch,definition,(singletonsswitch = (![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))).
% 15.11/15.26 thf(def_upairsetE,definition,(upairsetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))))).
% 15.11/15.26 thf(def_upairsetIL,definition,(upairsetIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))).
% 15.11/15.26 thf(def_upairsetIR,definition,(upairsetIR = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))).
% 15.11/15.26 thf(def_emptyE1,definition,(emptyE1 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))))).
% 15.11/15.26 thf(def_vacuousDall,definition,(vacuousDall = (![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))))).
% 15.11/15.26 thf(def_quantDeMorgan1,definition,(quantDeMorgan1 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))).
% 15.11/15.26 thf(def_quantDeMorgan2,definition,(quantDeMorgan2 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))).
% 15.11/15.26 thf(def_quantDeMorgan3,definition,(quantDeMorgan3 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))).
% 15.11/15.26 thf(def_quantDeMorgan4,definition,(quantDeMorgan4 = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))).
% 15.11/15.26 thf(def_prop2setI,definition,(prop2setI = (![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))))).
% 15.11/15.26 thf(def_prop2set2propI,definition,(prop2set2propI = (![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))))).
% 15.11/15.26 thf(def_notdexE,definition,(notdexE = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))).
% 15.11/15.26 thf(def_notdallE,definition,(notdallE = (![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))))).
% 15.11/15.26 thf(def_exuI1,definition,(exuI1 = (![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))))).
% 15.11/15.26 thf(def_exuI3,definition,(exuI3 = (![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))))).
% 15.11/15.26 thf(def_exuI2,definition,(exuI2 = (![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))))).
% 15.11/15.26 thf(def_inCongP,definition,(inCongP = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))))).
% 15.11/15.26 thf(def_in__Cong,definition,(in__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))))).
% 15.11/15.26 thf(def_exuE3u,definition,(exuE3u = (![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))))).
% 15.11/15.26 thf(def_exu__Cong,definition,(exu__Cong = (![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))))).
% 15.11/15.26 thf(def_emptyset__Cong,definition,(emptyset__Cong = (emptyset = emptyset))).
% 15.11/15.26 thf(def_setadjoin__Cong,definition,(setadjoin__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))))).
% 15.11/15.26 thf(def_powerset__Cong,definition,(powerset__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))))).
% 15.11/15.26 thf(def_setunion__Cong,definition,(setunion__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))))).
% 15.11/15.26 thf(def_omega__Cong,definition,(omega__Cong = (omega = omega))).
% 15.11/15.26 thf(def_exuEu,definition,(exuEu = (![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))))).
% 15.11/15.26 thf(def_descr__Cong,definition,(descr__Cong = (![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))))).
% 15.11/15.26 thf(def_dsetconstr__Cong,definition,(dsetconstr__Cong = (![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))))).
% 15.11/15.26 thf(def_subsetI1,definition,(subsetI1 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))))).
% 15.11/15.26 thf(def_eqimpsubset2,definition,(eqimpsubset2 = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))))).
% 15.11/15.26 thf(def_eqimpsubset1,definition,(eqimpsubset1 = (![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))))).
% 15.11/15.26 thf(def_subsetI2,definition,(subsetI2 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))))).
% 15.11/15.26 thf(def_emptysetsubset,definition,(emptysetsubset = ((!!) @ (subset @ emptyset)))).
% 15.11/15.26 thf(def_subsetE,definition,(subsetE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))))).
% 15.11/15.26 thf(def_subsetE2,definition,(subsetE2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))))).
% 15.11/15.26 thf(def_notsubsetI,definition,(notsubsetI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_notequalI1,definition,(notequalI1 = (![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))))).
% 15.11/15.26 thf(def_notequalI2,definition,(notequalI2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))))).
% 15.11/15.26 thf(def_subsetRefl,definition,(subsetRefl = (![X1:$i]:((subset @ X1) @ X1)))).
% 15.11/15.26 thf(def_subsetTrans,definition,(subsetTrans = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))))).
% 15.11/15.26 thf(def_setadjoinSub,definition,(setadjoinSub = (![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))))).
% 15.11/15.26 thf(def_setadjoinSub2,definition,(setadjoinSub2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))))).
% 15.11/15.26 thf(def_subset2powerset,definition,(subset2powerset = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))))).
% 15.11/15.26 thf(def_setextsub,definition,(setextsub = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))))).
% 15.11/15.26 thf(def_subsetemptysetimpeq,definition,(subsetemptysetimpeq = (![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))))).
% 15.11/15.26 thf(def_powersetI1,definition,(powersetI1 = (![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))))).
% 15.11/15.26 thf(def_powersetE1,definition,(powersetE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))))).
% 15.11/15.26 thf(def_inPowerset,definition,(inPowerset = (![X1:$i]:((in @ X1) @ (powerset @ X1))))).
% 15.11/15.26 thf(def_powersetsubset,definition,(powersetsubset = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))))).
% 15.11/15.26 thf(def_sepInPowerset,definition,(sepInPowerset = (![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))))).
% 15.11/15.26 thf(def_sepSubset,definition,(sepSubset = (![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))))).
% 15.11/15.26 thf(def_binunionIL,definition,(binunionIL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_upairset2IR,definition,(upairset2IR = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))).
% 15.11/15.26 thf(def_binunionIR,definition,(binunionIR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_binunionEcases,definition,(binunionEcases = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))))).
% 15.11/15.26 thf(def_binunionE,definition,(binunionE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))))).
% 15.11/15.26 thf(def_binunionLsub,definition,(binunionLsub = (![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))))).
% 15.11/15.26 thf(def_binunionRsub,definition,(binunionRsub = (![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))))).
% 15.11/15.26 thf(def_binintersectI,definition,(binintersectI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_binintersectSubset5,definition,(binintersectSubset5 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_binintersectEL,definition,(binintersectEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))))).
% 15.11/15.26 thf(def_binintersectLsub,definition,(binintersectLsub = (![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))))).
% 15.11/15.26 thf(def_binintersectSubset2,definition,(binintersectSubset2 = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))))).
% 15.11/15.26 thf(def_binintersectSubset3,definition,(binintersectSubset3 = (![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))))).
% 15.11/15.26 thf(def_binintersectER,definition,(binintersectER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))))).
% 15.11/15.26 thf(def_disjointsetsI1,definition,(disjointsetsI1 = (![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))))).
% 15.11/15.26 thf(def_binintersectRsub,definition,(binintersectRsub = (![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))))).
% 15.11/15.26 thf(def_binintersectSubset4,definition,(binintersectSubset4 = (![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))))).
% 15.11/15.26 thf(def_binintersectSubset1,definition,(binintersectSubset1 = (![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))))).
% 15.11/15.26 thf(def_bs114d,definition,(bs114d = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))))).
% 15.11/15.26 thf(def_setminusI,definition,(setminusI = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_setminusEL,definition,(setminusEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))))).
% 15.11/15.26 thf(def_setminusER,definition,(setminusER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))))).
% 15.11/15.26 thf(def_setminusSubset2,definition,(setminusSubset2 = (![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))))).
% 15.11/15.26 thf(def_setminusERneg,definition,(setminusERneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))))).
% 15.11/15.26 thf(def_setminusELneg,definition,(setminusELneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))))).
% 15.11/15.26 thf(def_setminusILneg,definition,(setminusILneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_setminusIRneg,definition,(setminusIRneg = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_setminusLsub,definition,(setminusLsub = (![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))))).
% 15.11/15.26 thf(def_setminusSubset1,definition,(setminusSubset1 = (![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))))).
% 15.11/15.26 thf(def_symdiffE,definition,(symdiffE = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))))).
% 15.11/15.26 thf(def_symdiffI1,definition,(symdiffI1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_symdiffI2,definition,(symdiffI2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_symdiffIneg1,definition,(symdiffIneg1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))))).
% 15.11/15.26 thf(def_symdiffIneg2,definition,(symdiffIneg2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))))).
% 15.11/15.26 thf(def_secondinupair,definition,(secondinupair = (![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))))).
% 15.11/15.26 thf(def_setukpairIL,definition,(setukpairIL = (![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))))).
% 15.11/15.26 thf(def_setukpairIR,definition,(setukpairIR = (![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))))).
% 15.11/15.26 thf(def_kpairiskpair,definition,(kpairiskpair = (![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))))).
% 15.11/15.26 thf(def_kpairp,definition,(kpairp = (![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))))).
% 15.11/15.26 thf(def_singletonsubset,definition,(singletonsubset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))))).
% 15.11/15.26 thf(def_singletoninpowerset,definition,(singletoninpowerset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))))).
% 15.11/15.26 thf(def_singletoninpowunion,definition,(singletoninpowunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_upairset2E,definition,(upairset2E = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))))).
% 15.11/15.26 thf(def_upairsubunion,definition,(upairsubunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_upairinpowunion,definition,(upairinpowunion = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))))).
% 15.11/15.26 thf(def_ubforcartprodlem1,definition,(ubforcartprodlem1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))))).
% 15.11/15.26 thf(def_ubforcartprodlem2,definition,(ubforcartprodlem2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))))).
% 15.11/15.26 thf(def_ubforcartprodlem3,definition,(ubforcartprodlem3 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))))).
% 15.11/15.26 thf(def_cartprodpairin,definition,(cartprodpairin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_cartprodmempair1,definition,(cartprodmempair1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))))).
% 15.11/15.26 thf(def_cartprodmempair,definition,(cartprodmempair = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))))).
% 15.11/15.26 thf(def_setunionE2,definition,(setunionE2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))))).
% 15.11/15.26 thf(def_setunionsingleton1,definition,(setunionsingleton1 = (![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)))).
% 15.11/15.26 thf(def_setunionsingleton2,definition,(setunionsingleton2 = (![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))))).
% 15.11/15.26 thf(def_setunionsingleton,definition,(setunionsingleton = (![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)))).
% 15.11/15.26 thf(def_singletonprop,definition,(singletonprop = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_ex1E1,definition,(ex1E1 = (![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))))).
% 15.11/15.26 thf(def_ex1I,definition,(ex1I = (![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))))).
% 15.11/15.26 thf(def_ex1I2,definition,(ex1I2 = (![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))))).
% 15.11/15.26 thf(def_singletonsuniq,definition,(singletonsuniq = (![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))))).
% 15.11/15.26 thf(def_setukpairinjL1,definition,(setukpairinjL1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))))).
% 15.11/15.26 thf(def_kfstsingleton,definition,(kfstsingleton = (![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))))).
% 15.11/15.26 thf(def_theprop,definition,(theprop = (![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))))).
% 15.11/15.26 thf(def_kfstpairEq,definition,(kfstpairEq = (![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))))).
% 15.11/15.26 thf(def_cartprodfstin,definition,(cartprodfstin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))))).
% 15.11/15.26 thf(def_setukpairinjL2,definition,(setukpairinjL2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))))).
% 15.11/15.26 thf(def_setukpairinjL,definition,(setukpairinjL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))))).
% 15.11/15.26 thf(def_setukpairinjR11,definition,(setukpairinjR11 = (![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))))).
% 15.11/15.26 thf(def_setukpairinjR12,definition,(setukpairinjR12 = (![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))))).
% 15.11/15.26 thf(def_setukpairinjR1,definition,(setukpairinjR1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))))).
% 15.11/15.26 thf(def_upairequniteq,definition,(upairequniteq = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))))).
% 15.11/15.26 thf(def_setukpairinjR2,definition,(setukpairinjR2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))))).
% 15.11/15.26 thf(def_setukpairinjR,definition,(setukpairinjR = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))))).
% 15.11/15.26 thf(def_ksndsingleton,definition,(ksndsingleton = (![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))))).
% 15.11/15.26 thf(def_ksndpairEq,definition,(ksndpairEq = (![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))))).
% 15.11/15.26 thf(def_kpairsurjEq,definition,(kpairsurjEq = (![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))))).
% 15.11/15.26 thf(def_cartprodsndin,definition,(cartprodsndin = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))))).
% 15.11/15.26 thf(def_cartprodpairmemEL,definition,(cartprodpairmemEL = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))))).
% 15.11/15.26 thf(def_cartprodpairmemER,definition,(cartprodpairmemER = (![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))))).
% 15.11/15.26 thf(def_cartprodmempaircEq,definition,(cartprodmempaircEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))))).
% 15.11/15.26 thf(def_cartprodfstpairEq,definition,(cartprodfstpairEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))))).
% 15.11/15.26 thf(def_cartprodsndpairEq,definition,(cartprodsndpairEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))))).
% 15.11/15.26 thf(def_cartprodpairsurjEq,definition,(cartprodpairsurjEq = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))))).
% 15.11/15.26 thf(def_dpsetconstrI,definition,(dpsetconstrI = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))))).
% 15.11/15.26 thf(def_dpsetconstrSub,definition,(dpsetconstrSub = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))))).
% 15.11/15.26 thf(def_setOfPairsIsBReln,definition,(setOfPairsIsBReln = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))))).
% 15.11/15.26 thf(def_dpsetconstrERa,definition,(dpsetconstrERa = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))))).
% 15.11/15.26 thf(def_dpsetconstrEL1,definition,(dpsetconstrEL1 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))))).
% 15.11/15.26 thf(def_dpsetconstrEL2,definition,(dpsetconstrEL2 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))))).
% 15.11/15.26 thf(def_dpsetconstrER,definition,(dpsetconstrER = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))).
% 15.11/15.26 thf(def_funcImageSingleton,definition,(funcImageSingleton = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))).
% 15.11/15.26 thf(def_apProp,definition,(apProp = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))))).
% 15.11/15.26 thf(def_app,definition,(app = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))))).
% 15.11/15.26 thf(def_infuncsetfunc,definition,(infuncsetfunc = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))))).
% 15.11/15.26 thf(def_ap2p,definition,(ap2p = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))))).
% 15.11/15.26 thf(def_funcinfuncset,definition,(funcinfuncset = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_lamProp,definition,(lamProp = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))))).
% 15.11/15.26 thf(def_lamp,definition,(lamp = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))))).
% 15.11/15.26 thf(def_lam2p,definition,(lam2p = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_brelnall1,definition,(brelnall1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))))).
% 15.11/15.26 thf(def_brelnall2,definition,(brelnall2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))))).
% 15.11/15.26 thf(def_ex1E2,definition,(ex1E2 = (![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))))).
% 15.11/15.26 thf(def_funcGraphProp1,definition,(funcGraphProp1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))))).
% 15.11/15.26 thf(def_funcGraphProp3,definition,(funcGraphProp3 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))))).
% 15.11/15.26 thf(def_funcGraphProp2,definition,(funcGraphProp2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))))).
% 15.11/15.26 thf(def_funcextLem,definition,(funcextLem = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))))).
% 15.11/15.26 thf(def_funcGraphProp4,definition,(funcGraphProp4 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))))).
% 15.11/15.26 thf(def_subbreln,definition,(subbreln = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))))).
% 15.11/15.26 thf(def_eqbreln,definition,(eqbreln = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))))).
% 15.11/15.26 thf(def_funcext,definition,(funcext = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))))).
% 15.11/15.26 thf(def_funcext2,definition,(funcext2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))))).
% 15.11/15.26 thf(def_ap2apEq1,definition,(ap2apEq1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))))).
% 15.11/15.26 thf(def_ap2apEq2,definition,(ap2apEq2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))))).
% 15.11/15.26 thf(def_beta1,definition,(beta1 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))))).
% 15.11/15.26 thf(def_eta1,definition,(eta1 = (![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))))).
% 15.11/15.26 thf(def_lam2lamEq,definition,(lam2lamEq = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))))).
% 15.11/15.26 thf(def_beta2,definition,(beta2 = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))))).
% 15.11/15.26 thf(def_eta2,definition,(eta2 = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))))).
% 15.11/15.26 thf(def_iffalseProp1,definition,(iffalseProp1 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))))).
% 15.11/15.26 thf(def_iffalseProp2,definition,(iffalseProp2 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))))).
% 15.11/15.26 thf(def_iftrueProp1,definition,(iftrueProp1 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))))).
% 15.11/15.26 thf(def_iftrueProp2,definition,(iftrueProp2 = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))))).
% 15.11/15.26 thf(def_ifSingleton,definition,(ifSingleton = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))))).
% 15.11/15.26 thf(def_ifp,definition,(ifp = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))))).
% 15.11/15.26 thf(def_theeq,definition,(theeq = (![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))))).
% 15.11/15.26 thf(def_iftrue,definition,(iftrue = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))))).
% 15.11/15.26 thf(def_iffalse,definition,(iffalse = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))))).
% 15.11/15.26 thf(def_iftrueorfalse,definition,(iftrueorfalse = (![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))))).
% 15.11/15.26 thf(def_binintersectT_lem,definition,(binintersectT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))))).
% 15.11/15.26 thf(def_binunionT_lem,definition,(binunionT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))))).
% 15.11/15.26 thf(def_powersetT_lem,definition,(powersetT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))))).
% 15.11/15.26 thf(def_setminusT_lem,definition,(setminusT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))))).
% 15.11/15.26 thf(def_complementT_lem,definition,(complementT_lem = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))))).
% 15.11/15.26 thf(def_setextT,definition,(setextT = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))))).
% 15.11/15.26 thf(def_subsetTI,definition,(subsetTI = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))))).
% 15.11/15.26 thf(def_powersetTI1,definition,(powersetTI1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))))).
% 15.11/15.26 thf(def_powersetTE1,definition,(powersetTE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))))).
% 15.11/15.26 thf(def_complementTI1,definition,(complementTI1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))))).
% 15.11/15.26 thf(def_complementTE1,definition,(complementTE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))))).
% 15.11/15.26 thf(def_binintersectTELcontra,definition,(binintersectTELcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_binintersectTERcontra,definition,(binintersectTERcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_contrasubsetT,definition,(contrasubsetT = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))))).
% 15.11/15.26 thf(def_contrasubsetT1,definition,(contrasubsetT1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))))).
% 15.11/15.26 thf(def_contrasubsetT2,definition,(contrasubsetT2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_contrasubsetT3,definition,(contrasubsetT3 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))))).
% 15.11/15.26 thf(def_doubleComplementI1,definition,(doubleComplementI1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))))).
% 15.11/15.26 thf(def_doubleComplementE1,definition,(doubleComplementE1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))))).
% 15.11/15.26 thf(def_doubleComplementSub1,definition,(doubleComplementSub1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_doubleComplementSub2,definition,(doubleComplementSub2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))))).
% 15.11/15.26 thf(def_doubleComplementEq,definition,(doubleComplementEq = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_complementTnotintersectT,definition,(complementTnotintersectT = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_complementImpComplementIntersect,definition,(complementImpComplementIntersect = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))))).
% 15.11/15.26 thf(def_complementSubsetComplementIntersect,definition,(complementSubsetComplementIntersect = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))))).
% 15.11/15.26 thf(def_complementInPowersetComplementIntersect,definition,(complementInPowersetComplementIntersect = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))).
% 15.11/15.26 thf(def_contraSubsetComplement,definition,(contraSubsetComplement = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))))).
% 15.11/15.26 thf(def_complementTcontraSubset,definition,(complementTcontraSubset = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_binunionTILcontra,definition,(binunionTILcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))))).
% 15.11/15.26 thf(def_binunionTIRcontra,definition,(binunionTIRcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))))).
% 15.11/15.26 thf(def_inIntersectImpInUnion,definition,(inIntersectImpInUnion = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))))).
% 15.11/15.26 thf(def_inIntersectImpInUnion2,definition,(inIntersectImpInUnion2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))))).
% 15.11/15.26 thf(def_inIntersectImpInIntersectUnions,definition,(inIntersectImpInIntersectUnions = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))))).
% 15.11/15.26 thf(def_intersectInPowersetIntersectUnions,definition,(intersectInPowersetIntersectUnions = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))).
% 15.11/15.26 thf(def_inComplementUnionImpNotIn1,definition,(inComplementUnionImpNotIn1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))))).
% 15.11/15.26 thf(def_inComplementUnionImpInComplement1,definition,(inComplementUnionImpInComplement1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))))).
% 15.11/15.26 thf(def_binunionTE,definition,(binunionTE = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))))).
% 15.11/15.26 thf(def_binunionTEcontra,definition,(binunionTEcontra = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))))).
% 15.11/15.26 thf(def_demorgan2a1,definition,(demorgan2a1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))))).
% 15.11/15.26 thf(def_complementUnionInPowersetComplement,definition,(complementUnionInPowersetComplement = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_demorgan2a2,definition,(demorgan2a2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))))).
% 15.11/15.26 thf(def_demorgan1a,definition,(demorgan1a = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))))).
% 15.11/15.26 thf(def_demorgan1b,definition,(demorgan1b = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))))).
% 15.11/15.26 thf(def_demorgan1,definition,(demorgan1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))))).
% 15.11/15.26 thf(def_demorgan2a,definition,(demorgan2a = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))))).
% 15.11/15.26 thf(def_demorgan2b2,definition,(demorgan2b2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_demorgan2b,definition,(demorgan2b = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))))).
% 15.11/15.26 thf(def_demorgan2,definition,(demorgan2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))))).
% 15.11/15.26 thf(def_woz13rule0,definition,(woz13rule0 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))))).
% 15.11/15.26 thf(def_woz13rule1,definition,(woz13rule1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))))).
% 15.11/15.26 thf(def_woz13rule2,definition,(woz13rule2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))))).
% 15.11/15.26 thf(def_woz13rule3,definition,(woz13rule3 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))))).
% 15.11/15.26 thf(def_woz13rule4,definition,(woz13rule4 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))))).
% 15.11/15.26 thf(def_woz1_1,definition,(woz1_1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))).
% 15.11/15.26 thf(def_woz1_2,definition,(woz1_2 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))))).
% 15.11/15.26 thf(def_woz1_3,definition,(woz1_3 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))).
% 15.11/15.26 thf(def_woz1_4,definition,(woz1_4 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_woz1_5,definition,(woz1_5 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_breln1all2,definition,(breln1all2 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))))).
% 15.11/15.26 thf(def_breln1SetBreln1,definition,(breln1SetBreln1 = (![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))))).
% 15.11/15.26 thf(def_choice2fnsingleton,definition,(choice2fnsingleton = (![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))))).
% 15.11/15.26 thf(def_setOfPairsIsBReln1,definition,(setOfPairsIsBReln1 = (![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))))).
% 15.11/15.26 thf(def_breln1all1,definition,(breln1all1 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))))).
% 15.11/15.26 thf(def_subbreln1,definition,(subbreln1 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))))).
% 15.11/15.26 thf(def_eqbreln1,definition,(eqbreln1 = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))))).
% 15.11/15.26 thf(def_breln1invprop,definition,(breln1invprop = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))))).
% 15.11/15.26 thf(def_breln1invI,definition,(breln1invI = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))))).
% 15.11/15.26 thf(def_breln1invE,definition,(breln1invE = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))))).
% 15.11/15.26 thf(def_breln1compprop,definition,(breln1compprop = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))))).
% 15.11/15.26 thf(def_breln1compI,definition,(breln1compI = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))))).
% 15.11/15.26 thf(def_breln1compE,definition,(breln1compE = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))))).
% 15.11/15.26 thf(def_breln1compEex,definition,(breln1compEex = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))))).
% 15.11/15.26 thf(def_breln1unionprop,definition,(breln1unionprop = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))))).
% 15.11/15.26 thf(def_breln1unionIL,definition,(breln1unionIL = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_breln1unionIR,definition,(breln1unionIR = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_breln1unionI,definition,(breln1unionI = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))))).
% 15.11/15.26 thf(def_breln1unionE,definition,(breln1unionE = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))))).
% 15.11/15.26 thf(def_breln1unionEcases,definition,(breln1unionEcases = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))))).
% 15.11/15.26 thf(def_breln1unionCommutes,definition,(breln1unionCommutes = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))))).
% 15.11/15.26 thf(def_woz2Ex,definition,(woz2Ex = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))))).
% 15.11/15.26 thf(def_woz2W,definition,(woz2W = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))).
% 15.11/15.26 thf(def_woz2A,definition,(woz2A = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))))).
% 15.11/15.26 thf(def_woz2B,definition,(woz2B = (![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))))).
% 15.11/15.26 thf(def_image1Ex,definition,(image1Ex = (![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))))).
% 15.11/15.26 thf(def_image1Ex1,definition,(image1Ex1 = (![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))))).
% 15.11/15.26 thf(def_image1Equiv,definition,(image1Equiv = (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))))).
% 15.17/15.27 thf(def_image1E,definition,(image1E = (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))))).
% 15.17/15.27 thf(def_image1I,definition,(image1I = (![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))))).
% 15.17/15.27 thf(def_injFuncInInjFuncSet,definition,(injFuncInInjFuncSet = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))))).
% 15.17/15.27 thf(def_injFuncSetFuncIn,definition,(injFuncSetFuncIn = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))))).
% 15.17/15.27 thf(def_injFuncSetFuncInj,definition,(injFuncSetFuncInj = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))))).
% 15.17/15.27 thf(def_surjFuncSetFuncIn,definition,(surjFuncSetFuncIn = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))))).
% 15.17/15.27 thf(def_surjFuncSetFuncSurj,definition,(surjFuncSetFuncSurj = (![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))))).
% 15.17/15.27 thf(def_leftInvIsSurj,definition,(leftInvIsSurj = (![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))))).
% 15.17/15.27 thf(def_surjCantorThm,definition,(surjCantorThm = (![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))))).
% 15.17/15.27 thf(def_foundation2,definition,(foundation2 = (![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))))).
% 15.17/15.27 thf(def_notinself,definition,(notinself = (![X1:$i]:(~(((in @ X1) @ X1)))))).
% 15.17/15.27 thf(def_notinself2,definition,(notinself2 = (![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))))).
% 15.17/15.27 thf(def_omegaSp,definition,(omegaSp = (![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))))).
% 15.17/15.27 thf(def_omegaSclos,definition,(omegaSclos = (![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))))).
% 15.17/15.27 thf(def_peano0notS,definition,(peano0notS = (![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))))).
% 15.17/15.27 thf(def_peanoSinj,definition,(peanoSinj = (![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))))).
% 15.17/15.27 thf(def_transitiveset,definition,(transitiveset = (^[X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1)))))).
% 15.17/15.27 thf(def_transitivesetOp1,definition,(transitivesetOp1 = (![X1:$i]:((transitiveset @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))))).
% 15.17/15.27 thf(binintTransitive,conjecture,((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))).
% 15.17/15.27 thf(h0,negated_conjecture,(~(((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => 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X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),inference(assume_negation,[status(cth)],[binintTransitive])).
% 15.17/15.28 thf(h1,assumption,(![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) = ((in @ X3) @ X2))) => (X1 = X2)))),introduced(assumption,[])).
% 15.17/15.28 thf(h2,assumption,(~(((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.28 thf(h3,assumption,(![X1:$i]:(~(((in @ X1) @ emptyset)))),introduced(assumption,[])).
% 15.17/15.28 thf(h4,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin 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emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.29 thf(h5,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) = ((~((X3 = X1))) => ((in @ X3) @ X2)))))),introduced(assumption,[])).
% 15.17/15.29 thf(h6,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in 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X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ 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@ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) 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(powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.29 thf(h7,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) = (![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1)))))),introduced(assumption,[])).
% 15.17/15.29 thf(h8,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))) => (((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.29 thf(h9,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) = (~((![X3:$i]:(((in @ X2) @ X3) => (~(((in @ X3) @ X1)))))))))),introduced(assumption,[])).
% 15.17/15.29 thf(h10,assumption,(~((((in @ emptyset) @ omega) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ 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=> (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ 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((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.30 thf(h11,assumption,((in @ emptyset) @ omega),introduced(assumption,[])).
% 15.17/15.30 thf(h12,assumption,(~(((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.30 thf(h13,assumption,(![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))),introduced(assumption,[])).
% 15.17/15.30 thf(h14,assumption,(~(((![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))) => ((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ 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@ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.31 thf(h15,assumption,(![X1:$i]:((~((((in @ emptyset) @ X1) => (~((![X2:$i]:((~((((in @ X2) @ omega) => (~(((in @ X2) @ X1)))))) => ((in @ ((setadjoin @ X2) @ X2)) @ X1)))))))) => (![X2:$i]:(((in @ X2) @ omega) => ((in @ X2) @ X1))))),introduced(assumption,[])).
% 15.17/15.31 thf(h16,assumption,(~(((![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = 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emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin 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emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ 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((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.31 thf(h17,assumption,(![X1:$i>$i>$o]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => (exu @ (X1 @ X3)))) => (~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X1 @ X5) @ X4))))))))))))))))),introduced(assumption,[])).
% 15.17/15.31 thf(h18,assumption,(~(((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))) => ((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ 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=> ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => 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X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.32 thf(h19,assumption,(![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ X1))))))))))))),introduced(assumption,[])).
% 15.17/15.32 thf(h20,assumption,(~(((![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))) => ((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin 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@ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.32 thf(h21,assumption,(![X1:$i]:(~((![X2:$i]:((~(((~(((![X3:$i]:(((in @ X3) @ X2) => (![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X1) => (~(((in @ X4) @ X1)))))) => ((![X5:$i]:(((in @ X5) @ X2) => (((in @ X3) @ X5) = ((in @ X4) @ X5)))) => (X3 = X4)))))))))) => (~((![X3:$i]:(![X4:$i]:((~((((in @ X3) @ X2) => (~(((in @ X4) @ X2)))))) => ((~((![X5:$i]:(((in @ X5) @ X3) => ((in @ X5) @ X4))))) => (![X5:$i]:(((in @ X5) @ X4) => ((in @ X5) @ X3)))))))))))) => (~((![X3:$i]:((~(((![X4:$i]:(((in @ X4) @ X3) => ((in @ X4) @ X1))) => (![X4:$i]:(~(((in @ X4) @ X3))))))) => (~((![X4:$i]:(![X5:$i]:((~(((~((((in @ X4) @ X2) => (~(((in @ X5) @ X3)))))) => (~((![X6:$i]:(((in @ X6) @ X4) => (~(((in @ X6) @ X3)))))))))) => (~((![X6:$i]:(((in @ X6) @ X2) => ((~((![X7:$i]:(((in @ X7) @ X6) => ((in @ X7) @ X4))))) => ((in @ X5) @ X6)))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.32 thf(h22,assumption,(~(((![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.33 thf(h23,assumption,(![X1:$i>$o]:((exu @ X1) => (X1 @ (descr @ X1)))),introduced(assumption,[])).
% 15.17/15.33 thf(h24,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ 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(powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.33 thf(h25,assumption,(![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((in @ X3) @ ((dsetconstr @ X1) @ X2))))))),introduced(assumption,[])).
% 15.17/15.33 thf(h26,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ 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(![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => 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=> (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => 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=> ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.33 thf(h27,assumption,(![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => ((in @ X3) @ X1))))),introduced(assumption,[])).
% 15.17/15.33 thf(h28,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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=> (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ 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X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.34 thf(h29,assumption,(![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ ((dsetconstr @ X1) @ X2)) => (X2 @ X3))))),introduced(assumption,[])).
% 15.17/15.34 thf(h30,assumption,(~(((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))) => ((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => 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((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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(((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ 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X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => 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((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.35 thf(h31,assumption,(![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))))),introduced(assumption,[])).
% 15.17/15.35 thf(h32,assumption,(~(((![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))) => ((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ 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((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => 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@ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.35 thf(h33,assumption,(![X1:$o]:(![X2:$i]:(((in @ X2) @ (prop2set @ X1)) => X1))),introduced(assumption,[])).
% 15.17/15.35 thf(h34,assumption,(~(((![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = 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@ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => 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X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ 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emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => 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((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => 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((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.35 thf(h35,assumption,(![X1:$i]:(((in @ X1) @ emptyset) => (![X2:$o]:X2))),introduced(assumption,[])).
% 15.17/15.35 thf(h36,assumption,(~(((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.35 thf(h37,assumption,(~(((![X1:$i]:(~(((in @ X1) @ emptyset)))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ 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=> (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.36 thf(h38,assumption,(~(((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.36 thf(h39,assumption,(![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((X1 @ X2)))))))),introduced(assumption,[])).
% 15.17/15.36 thf(h40,assumption,(~(((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.17/15.37 thf(h41,assumption,(![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => (X1 = X2))))),introduced(assumption,[])).
% 15.17/15.37 thf(h42,assumption,(~(((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.37 thf(h43,assumption,(![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))),introduced(assumption,[])).
% 15.28/15.37 thf(h44,assumption,(~(((![X1:$i]:((![X2:$i]:(~(((in @ X2) @ X1)))) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ 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=> (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.38 thf(h45,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ 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(kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ 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X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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(![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.38 thf(h46,assumption,(![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((dsetconstr @ X1) @ X2)) = (X2 @ X3)))))),introduced(assumption,[])).
% 15.28/15.38 thf(h47,assumption,(~(((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => 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X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ 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((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.38 thf(h48,assumption,(![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))),introduced(assumption,[])).
% 15.28/15.38 thf(h49,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))) => ((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ 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@ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) 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(~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ 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X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.39 thf(h50,assumption,(![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => (nonempty @ ((dsetconstr @ X1) @ X2))))))),introduced(assumption,[])).
% 15.28/15.39 thf(h51,assumption,(~(((![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ 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=> (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.39 thf(h52,assumption,(![X1:$i]:((~((![X2:$i]:(~(((in @ X2) @ X1)))))) => (nonempty @ X1))),introduced(assumption,[])).
% 15.28/15.39 thf(h53,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))) => (((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.28/15.39 thf(h54,assumption,(![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ X2)))),introduced(assumption,[])).
% 15.28/15.39 thf(h55,assumption,(~((((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => 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((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => 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@ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.40 thf(h56,assumption,((in @ emptyset) @ ((setadjoin @ emptyset) @ emptyset)),introduced(assumption,[])).
% 15.31/15.40 thf(h57,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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(powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.40 thf(h58,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((setadjoin @ X1) @ X2)))))),introduced(assumption,[])).
% 15.31/15.40 thf(h59,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.40 thf(h60,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => (![X4:$o]:(((X3 = X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))),introduced(assumption,[])).
% 15.31/15.40 thf(h61,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => 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@ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.41 thf(h62,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ X2)) => ((~((X3 = X1))) => ((in @ X3) @ X2)))))),introduced(assumption,[])).
% 15.31/15.41 thf(h63,assumption,(~(((![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.41 thf(h64,assumption,(![X1:$i]:(((dsetconstr @ X1) @ (^[X2:$i]:(~($false)))) = X1)),introduced(assumption,[])).
% 15.31/15.41 thf(h65,assumption,(~(((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ 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=> (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.42 thf(h66,assumption,(![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ X1))) => ((in @ X2) @ (powerset @ X1))))),introduced(assumption,[])).
% 15.31/15.42 thf(h67,assumption,(~(((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin 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X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ 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((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.42 thf(h68,assumption,(![X1:$i]:((in @ emptyset) @ (powerset @ X1))),introduced(assumption,[])).
% 15.31/15.42 thf(h69,assumption,(~(((![X1:$i]:((in @ emptyset) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ 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X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.31/15.42 thf(h70,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => 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=> (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => 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=> ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.43 thf(h71,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ (powerset @ X1)) => (((in @ X3) @ X2) => ((in @ X3) @ X1)))))),introduced(assumption,[])).
% 15.34/15.43 thf(h72,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.43 thf(h73,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X2) @ X3) => (((in @ X3) @ X1) => ((in @ X2) @ (setunion @ X1))))))),introduced(assumption,[])).
% 15.34/15.43 thf(h74,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => 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@ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) 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X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.43 thf(h75,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (![X3:$o]:((![X4:$i]:(((in @ X2) @ X4) => (((in @ X4) @ X1) => X3))) => X3))))),introduced(assumption,[])).
% 15.34/15.43 thf(h76,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))) => ((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.43 thf(h77,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ X2) @ (powerset @ (setunion @ X1)))))),introduced(assumption,[])).
% 15.34/15.43 thf(h78,assumption,(~(((![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.44 thf(h79,assumption,(![X1:$i>$o]:((exu @ X1) => (~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))))),introduced(assumption,[])).
% 15.34/15.44 thf(h80,assumption,(~(((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(~(((in @ X2) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ 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(![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ 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=> ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => 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((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ 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(![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.44 thf(h81,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => 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X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = 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@ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = 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((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.44 thf(h82,assumption,(![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))),introduced(assumption,[])).
% 15.34/15.44 thf(h83,assumption,(~(((![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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@ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.45 thf(h84,assumption,(![X1:$i]:(![X2:$i]:((~((X1 = X2))) => (~(((in @ X2) @ ((setadjoin @ X1) @ emptyset))))))),introduced(assumption,[])).
% 15.34/15.45 thf(h85,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => 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=> (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.45 thf(h86,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => ((in @ X1) @ ((setadjoin @ X2) @ emptyset))))),introduced(assumption,[])).
% 15.34/15.45 thf(h87,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin 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@ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.45 thf(h88,assumption,(![X1:$i]:(![X2:$i]:(((in @ X1) @ ((setadjoin @ X2) @ emptyset)) => ((in @ X2) @ ((setadjoin @ X1) @ emptyset))))),introduced(assumption,[])).
% 15.34/15.45 thf(h89,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.46 thf(h90,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))),introduced(assumption,[])).
% 15.34/15.46 thf(h91,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.46 thf(h92,assumption,(![X1:$i]:(![X2:$i]:((in @ X1) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))),introduced(assumption,[])).
% 15.34/15.46 thf(h93,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => 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@ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ 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(![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ 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((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.46 thf(h94,assumption,(![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))),introduced(assumption,[])).
% 15.34/15.46 thf(h95,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))) => ((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => 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((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => 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@ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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@ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.47 thf(h96,assumption,(![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (~((((dsetconstr @ X1) @ X2) = emptyset)))))),introduced(assumption,[])).
% 15.34/15.47 thf(h97,assumption,(~(((![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.47 thf(h98,assumption,(![X1:$i>$o]:(![X2:$i]:(((in @ X2) @ emptyset) => (X1 @ X2)))),introduced(assumption,[])).
% 15.34/15.47 thf(h99,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ 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@ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.47 thf(h100,assumption,(![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))),introduced(assumption,[])).
% 15.34/15.47 thf(h101,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = 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X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => 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((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.47 thf(h102,assumption,(![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))),introduced(assumption,[])).
% 15.34/15.47 thf(h103,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ 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((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ 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=> ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => 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((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.48 thf(h104,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => 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(powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.48 thf(h105,assumption,(~(((![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))) => ((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ 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X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.48 thf(h106,assumption,(![X1:$o]:(X1 => ((in @ emptyset) @ (prop2set @ X1)))),introduced(assumption,[])).
% 15.34/15.48 thf(h107,assumption,(~(((![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => 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@ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.49 thf(h108,assumption,(![X1:$o]:(X1 => (set2prop @ (prop2set @ X1)))),introduced(assumption,[])).
% 15.34/15.49 thf(h109,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))) => (![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))) => ((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.49 thf(h110,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3))))) => (~((![X3:$i]:(((in @ X3) @ X1) => (X2 @ X3)))))))) => ((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ 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X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.34/15.49 thf(h111,assumption,(~(((![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ 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=> (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ 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X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.50 thf(h112,assumption,(![X1:$i>$o]:((~((![X2:$i]:((X1 @ X2) => (~((![X3:$i]:((X1 @ X3) => (X2 = X3))))))))) => (exu @ X1))),introduced(assumption,[])).
% 15.43/15.50 thf(h113,assumption,(~(((![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))) => ((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in 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((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 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X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.50 thf(h114,assumption,(![X1:$i>$o]:((~((![X2:$i]:(~((X1 @ X2)))))) => ((![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))) => (exu @ X1)))),introduced(assumption,[])).
% 15.43/15.50 thf(h115,assumption,(~(((![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => 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@ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.50 thf(h116,assumption,(![X1:$i>$o]:((~((![X2:$i]:(~((![X3:$i]:((X1 @ X3) = (X3 = X2)))))))) => (exu @ X1))),introduced(assumption,[])).
% 15.43/15.50 thf(h117,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ 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((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h118,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) => ((in @ X4) @ X2)))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h119,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin 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@ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h120,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((in @ X3) @ X1) = ((in @ X4) @ X2)))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h121,assumption,(~(((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ 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(![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) 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X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ 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@ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h122,assumption,(![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h123,assumption,(~(((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))) => ((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.51 thf(h124,assumption,(![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) = (exu @ X2))))),introduced(assumption,[])).
% 15.43/15.51 thf(h125,assumption,(~(((emptyset = emptyset) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => 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@ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) 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(![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ 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X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ 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((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ 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((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.52 thf(h126,assumption,(emptyset = emptyset),introduced(assumption,[])).
% 15.43/15.52 thf(h127,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.52 thf(h128,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i]:(![X4:$i]:((X3 = X4) => (((setadjoin @ X1) @ X3) = ((setadjoin @ X2) @ X4)))))))),introduced(assumption,[])).
% 15.43/15.52 thf(h129,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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=> (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.52 thf(h130,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => ((powerset @ X1) = (powerset @ X2))))),introduced(assumption,[])).
% 15.43/15.52 thf(h131,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))) => ((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => 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@ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ 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X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.53 thf(h132,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => ((setunion @ X1) = (setunion @ X2))))),introduced(assumption,[])).
% 15.43/15.53 thf(h133,assumption,(~(((omega = omega) => ((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ 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((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ 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X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.53 thf(h134,assumption,(omega = omega),introduced(assumption,[])).
% 15.43/15.53 thf(h135,assumption,(~(((![X1:$i>$o]:((exu @ X1) => (![X2:$i]:(![X3:$i]:((X1 @ X2) => ((X1 @ X3) => (X2 = X3))))))) => ((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.53 thf(h136,assumption,(~(((![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ 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((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.53 thf(h137,assumption,(![X1:$i>$o]:(![X2:$i>$o]:((![X3:$i]:(![X4:$i]:((X3 = X4) => ((X1 @ X3) = (X2 @ X4))))) => ((exu @ X1) => ((exu @ X2) => ((descr @ X1) = (descr @ X2))))))),introduced(assumption,[])).
% 15.43/15.54 thf(h138,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.54 thf(h139,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => (![X3:$i>$o]:(![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => ((X5 = X6) => ((X3 @ X5) = (X4 @ X6))))))) => (((dsetconstr @ X1) @ X3) = ((dsetconstr @ X2) @ X4)))))))),introduced(assumption,[])).
% 15.43/15.54 thf(h140,assumption,(~(((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) 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@ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.54 thf(h141,assumption,(![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))),introduced(assumption,[])).
% 15.43/15.54 thf(h142,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ 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X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.54 thf(h143,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X2) @ X1)))),introduced(assumption,[])).
% 15.43/15.54 thf(h144,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => 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@ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) 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(![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ 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X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ 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((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ 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((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.55 thf(h145,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => ((subset @ X1) @ X2)))),introduced(assumption,[])).
% 15.43/15.55 thf(h146,assumption,(~(((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ X2))) => ((subset @ X1) @ X2)))) => (((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ 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@ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.55 thf(h147,assumption,(~((((!!) @ (subset @ emptyset)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.55 thf(h148,assumption,((!!) @ (subset @ emptyset)),introduced(assumption,[])).
% 15.43/15.55 thf(h149,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ 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(![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.56 thf(h150,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))),introduced(assumption,[])).
% 15.43/15.56 thf(h151,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.56 thf(h152,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))),introduced(assumption,[])).
% 15.43/15.56 thf(h153,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.56 thf(h154,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~(((subset @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.43/15.56 thf(h155,assumption,(~(((![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ 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(((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.43/15.57 thf(h156,assumption,(![X1:$i]:(![X2:$i]:((~(((subset @ X1) @ X2))) => (~((X1 = X2)))))),introduced(assumption,[])).
% 15.43/15.57 thf(h157,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))) => ((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => 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=> (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.57 thf(h158,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => (~((X1 = X2)))))))),introduced(assumption,[])).
% 15.51/15.57 thf(h159,assumption,(~(((![X1:$i]:((subset @ X1) @ X1)) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => 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@ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.57 thf(h160,assumption,(![X1:$i]:((subset @ X1) @ X1)),introduced(assumption,[])).
% 15.51/15.57 thf(h161,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => 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@ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.57 thf(h162,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X3) => ((subset @ X1) @ X3)))))),introduced(assumption,[])).
% 15.51/15.57 thf(h163,assumption,(~(((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ 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=> (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.58 thf(h164,assumption,(![X1:$i]:(![X2:$i]:((subset @ X2) @ ((setadjoin @ X1) @ X2)))),introduced(assumption,[])).
% 15.51/15.58 thf(h165,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ 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=> (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.58 thf(h166,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X1) @ X3) => ((subset @ X1) @ ((setadjoin @ X2) @ X3)))))),introduced(assumption,[])).
% 15.51/15.58 thf(h167,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => 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X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => 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X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.58 thf(h168,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((in @ X1) @ (powerset @ X2))))),introduced(assumption,[])).
% 15.51/15.58 thf(h169,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))) => ((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.59 thf(h170,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((subset @ X2) @ X1) => (X1 = X2))))),introduced(assumption,[])).
% 15.51/15.59 thf(h171,assumption,(~(((![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => 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X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) 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((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.59 thf(h172,assumption,(![X1:$i]:(((subset @ X1) @ emptyset) => (X1 = emptyset))),introduced(assumption,[])).
% 15.51/15.59 thf(h173,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.59 thf(h174,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => ((in @ X2) @ (powerset @ X1))))),introduced(assumption,[])).
% 15.51/15.59 thf(h175,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))) => ((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.59 thf(h176,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ X1)))),introduced(assumption,[])).
% 15.51/15.59 thf(h177,assumption,(~(((![X1:$i]:((in @ X1) @ (powerset @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ 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X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln 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((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ 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@ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.60 thf(h178,assumption,(![X1:$i]:((in @ X1) @ (powerset @ X1))),introduced(assumption,[])).
% 15.51/15.60 thf(h179,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))) => ((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => 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((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ 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@ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.60 thf(h180,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => ((subset @ (powerset @ X1)) @ (powerset @ X2))))),introduced(assumption,[])).
% 15.51/15.60 thf(h181,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))) => ((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => 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((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.60 thf(h182,assumption,(![X1:$i]:(![X2:$i>$o]:((in @ ((dsetconstr @ X1) @ X2)) @ (powerset @ X1)))),introduced(assumption,[])).
% 15.51/15.60 thf(h183,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ 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(powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.61 thf(h184,assumption,(![X1:$i]:(![X2:$i>$o]:((subset @ ((dsetconstr @ X1) @ X2)) @ X1))),introduced(assumption,[])).
% 15.51/15.61 thf(h185,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.61 thf(h186,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ X3) @ ((binunion @ X1) @ X2)))))),introduced(assumption,[])).
% 15.51/15.61 thf(h187,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => 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=> (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.61 thf(h188,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ 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=> (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ 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(powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.62 thf(h189,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => ((in @ X3) @ ((binunion @ X1) @ X2)))))),introduced(assumption,[])).
% 15.51/15.62 thf(h190,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ 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X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.62 thf(h191,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$o]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((((in @ X3) @ X1) => X4) => ((((in @ X3) @ X2) => X4) => X4))))))),introduced(assumption,[])).
% 15.51/15.62 thf(h192,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.62 thf(h193,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binunion @ X1) @ X2)) => ((~(((in @ X3) @ X1))) => ((in @ X3) @ X2)))))),introduced(assumption,[])).
% 15.51/15.62 thf(h194,assumption,(~(((![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.62 thf(h195,assumption,(![X1:$i]:(![X2:$i]:((subset @ X1) @ ((binunion @ X1) @ X2)))),introduced(assumption,[])).
% 15.51/15.62 thf(h196,assumption,(~(((![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.63 thf(h197,assumption,(![X1:$i]:(![X2:$i]:((subset @ X2) @ ((binunion @ X1) @ X2)))),introduced(assumption,[])).
% 15.51/15.63 thf(h198,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => 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emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = 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((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.63 thf(h199,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((binintersect @ X1) @ X2))))))),introduced(assumption,[])).
% 15.51/15.63 thf(h200,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ 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((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.63 thf(h201,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((subset @ X3) @ X1) => (((subset @ X3) @ X2) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))),introduced(assumption,[])).
% 15.51/15.63 thf(h202,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ 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X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.63 thf(h203,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X1))))),introduced(assumption,[])).
% 15.51/15.63 thf(h204,assumption,(~(((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin 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=> (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.64 thf(h205,assumption,(![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X1))),introduced(assumption,[])).
% 15.51/15.64 thf(h206,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ 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@ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.64 thf(h207,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((binintersect @ X1) @ X2) = X1)))),introduced(assumption,[])).
% 15.51/15.64 thf(h208,assumption,(~(((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.64 thf(h209,assumption,(![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X2) => ((subset @ X2) @ X1)))),introduced(assumption,[])).
% 15.51/15.64 thf(h210,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))) => ((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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(![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ 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@ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) 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((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.65 thf(h211,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((in @ X3) @ X2))))),introduced(assumption,[])).
% 15.51/15.65 thf(h212,assumption,(~(((![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.65 thf(h213,assumption,(![X1:$i]:(![X2:$i]:((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X3) @ X2))))) => (((binintersect @ X1) @ X2) = emptyset)))),introduced(assumption,[])).
% 15.51/15.65 thf(h214,assumption,(~(((![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))) => ((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.65 thf(h215,assumption,(![X1:$i]:(![X2:$i]:((subset @ ((binintersect @ X1) @ X2)) @ X2))),introduced(assumption,[])).
% 15.51/15.65 thf(h216,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))) => ((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ 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((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ 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X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.65 thf(h217,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X2) @ X1) => (((binintersect @ X1) @ X2) = X2)))),introduced(assumption,[])).
% 15.51/15.65 thf(h218,assumption,(~(((![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => 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= X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => 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((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ 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((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.65 thf(h219,assumption,(![X1:$i]:(![X2:$i]:((((binintersect @ X1) @ X2) = X1) => ((subset @ X1) @ X2)))),introduced(assumption,[])).
% 15.51/15.65 thf(h220,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => 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@ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.66 thf(h221,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((binintersect @ X1) @ ((binunion @ X2) @ X3)) = ((binunion @ ((binintersect @ X1) @ X2)) @ ((binintersect @ X1) @ X3)))))),introduced(assumption,[])).
% 15.51/15.66 thf(h222,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ 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X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => 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X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.66 thf(h223,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((setminus @ X1) @ X2))))))),introduced(assumption,[])).
% 15.51/15.66 thf(h224,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ 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(![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.67 thf(h225,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => ((in @ X3) @ X1))))),introduced(assumption,[])).
% 15.51/15.67 thf(h226,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.67 thf(h227,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setminus @ X1) @ X2)) => (~(((in @ X3) @ X2))))))),introduced(assumption,[])).
% 15.51/15.67 thf(h228,assumption,(~(((![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ 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(((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.51/15.67 thf(h229,assumption,(![X1:$i]:(![X2:$i]:(((subset @ X1) @ X2) => (((setminus @ X1) @ X2) = emptyset)))),introduced(assumption,[])).
% 15.51/15.67 thf(h230,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ 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(![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in 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X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.67 thf(h231,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => (((in @ X3) @ X1) => ((in @ X3) @ X2)))))),introduced(assumption,[])).
% 15.62/15.67 thf(h232,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ 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@ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.67 thf(h233,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ X1)))))))),introduced(assumption,[])).
% 15.62/15.67 thf(h234,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ 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=> (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.68 thf(h235,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.62/15.68 thf(h236,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.68 thf(h237,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.62/15.68 thf(h238,assumption,(~(((![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))) => ((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => 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((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ 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@ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.68 thf(h239,assumption,(![X1:$i]:(![X2:$i]:((subset @ ((setminus @ X1) @ X2)) @ X1))),introduced(assumption,[])).
% 15.62/15.68 thf(h240,assumption,(~(((![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h241,assumption,(![X1:$i]:(![X2:$i]:((((setminus @ X1) @ X2) = emptyset) => ((subset @ X1) @ X2)))),introduced(assumption,[])).
% 15.62/15.69 thf(h242,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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(powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h243,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((symdiff @ X1) @ X2)) => (![X4:$o]:((((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => X4)) => (((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => X4)) => X4))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h244,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ 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((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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(powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h245,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ X2))) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h246,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => 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@ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => 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X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => 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(![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h247,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => (((in @ X3) @ X2) => ((in @ X3) @ ((symdiff @ X1) @ X2))))))),introduced(assumption,[])).
% 15.62/15.69 thf(h248,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) 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@ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.70 thf(h249,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))),introduced(assumption,[])).
% 15.64/15.70 thf(h250,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.70 thf(h251,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((~(((in @ X3) @ X1))) => ((~(((in @ X3) @ X2))) => (~(((in @ X3) @ ((symdiff @ X1) @ X2))))))))),introduced(assumption,[])).
% 15.64/15.70 thf(h252,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X2) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.70 thf(h253,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ 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((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ 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@ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.71 thf(h254,assumption,(![X1:$i]:(![X2:$i]:((in @ X1) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))),introduced(assumption,[])).
% 15.64/15.71 thf(h255,assumption,(~(((![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ 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((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.71 thf(h256,assumption,(![X1:$i]:(![X2:$i]:((in @ X2) @ (setunion @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)))))),introduced(assumption,[])).
% 15.64/15.71 thf(h257,assumption,(~(((![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = 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=> (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.71 thf(h258,assumption,(![X1:$i]:(![X2:$i]:(iskpair @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))))),introduced(assumption,[])).
% 15.64/15.71 thf(h259,assumption,(~(((![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ 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X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ 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((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ 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((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.71 thf(h260,assumption,(![X1:$i]:(![X2:$i]:(iskpair @ ((kpair @ X1) @ X2)))),introduced(assumption,[])).
% 15.64/15.71 thf(h261,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => 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(![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.72 thf(h262,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((subset @ ((setadjoin @ X2) @ emptyset)) @ X1)))),introduced(assumption,[])).
% 15.64/15.72 thf(h263,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) 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=> (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.64/15.72 thf(h264,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ X1) => ((in @ ((setadjoin @ X2) @ emptyset)) @ (powerset @ X1))))),introduced(assumption,[])).
% 15.64/15.72 thf(h265,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ 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((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.72 thf(h266,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => ((in @ ((setadjoin @ X3) @ emptyset)) @ (powerset @ ((binunion @ X1) @ X2))))))),introduced(assumption,[])).
% 15.67/15.72 thf(h267,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) => ((~((X3 = X1))) => (X3 = X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ 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((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => 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X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
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X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ 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X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.72 thf(h269,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ ((binunion @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.67/15.72 thf(h270,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => 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@ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h271,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h272,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => 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=> (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h273,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((subset @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ ((binunion @ X1) @ X2))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h274,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ 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(![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h275,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h276,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ 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(^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h277,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ (powerset @ (powerset @ ((binunion @ X1) @ X2)))))))))),introduced(assumption,[])).
% 15.67/15.73 thf(h278,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => 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X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.74 thf(h279,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.67/15.74 thf(h280,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.67/15.74 thf(h281,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (~((X3 = ((kpair @ X4) @ X5)))))))))))))),introduced(assumption,[])).
% 15.67/15.74 thf(h282,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.74 thf(h283,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (iskpair @ X3))))),introduced(assumption,[])).
% 15.70/15.74 thf(h284,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))) => ((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.74 thf(h285,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (setunion @ X1)) => (~((![X3:$i]:(((in @ X3) @ X1) => (~(((in @ X2) @ X3)))))))))),introduced(assumption,[])).
% 15.70/15.74 thf(h286,assumption,(~(((![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)) => ((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ 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X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.75 thf(h287,assumption,(![X1:$i]:((subset @ (setunion @ ((setadjoin @ X1) @ emptyset))) @ X1)),introduced(assumption,[])).
% 15.70/15.75 thf(h288,assumption,(~(((![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))) => ((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.75 thf(h289,assumption,(![X1:$i]:((subset @ X1) @ (setunion @ ((setadjoin @ X1) @ emptyset)))),introduced(assumption,[])).
% 15.70/15.75 thf(h290,assumption,(~(((![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.75 thf(h291,assumption,(![X1:$i]:((setunion @ ((setadjoin @ X1) @ emptyset)) = X1)),introduced(assumption,[])).
% 15.70/15.75 thf(h292,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) 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(![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.75 thf(h293,assumption,(![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => (singleton @ ((dsetconstr @ X1) @ X2)))))),introduced(assumption,[])).
% 15.70/15.75 thf(h294,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))) => ((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ 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X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.76 thf(h295,assumption,(![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3)))))))))),introduced(assumption,[])).
% 15.70/15.76 thf(h296,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.76 thf(h297,assumption,(![X1:$i]:(![X2:$i>$o]:(![X3:$i]:(((in @ X3) @ X1) => ((X2 @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X4) => (X4 = X3)))) => ((ex1 @ X1) @ X2))))))),introduced(assumption,[])).
% 15.70/15.76 thf(h298,assumption,(~(((![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => 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(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.76 thf(h299,assumption,(![X1:$i]:(![X2:$i>$o]:((![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4))))))) => ((~((![X3:$i]:(((in @ X3) @ X1) => (~((X2 @ X3))))))) => ((ex1 @ X1) @ X2))))),introduced(assumption,[])).
% 15.70/15.76 thf(h300,assumption,(~(((![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ 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@ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => 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@ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.77 thf(h301,assumption,(![X1:$i]:(![X2:$i]:((((setadjoin @ X1) @ emptyset) = ((setadjoin @ X2) @ emptyset)) => (X1 = X2)))),introduced(assumption,[])).
% 15.70/15.77 thf(h302,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) 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X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.77 thf(h303,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset))) => (X1 = X3))))),introduced(assumption,[])).
% 15.70/15.77 thf(h304,assumption,(~(((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))) => ((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.77 thf(h305,assumption,(![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:((in @ ((setadjoin @ X2) @ emptyset)) @ X1)))))),introduced(assumption,[])).
% 15.70/15.77 thf(h306,assumption,(~(((![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))) => ((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => 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(![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ 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X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ 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=> (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.77 thf(h307,assumption,(![X1:$i]:((singleton @ X1) => ((in @ (setunion @ X1)) @ X1))),introduced(assumption,[])).
% 15.70/15.77 thf(h308,assumption,(~(((![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ 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((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ 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X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => 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X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.78 thf(h309,assumption,(![X1:$i]:(![X2:$i]:((kfst @ ((kpair @ X1) @ X2)) = X1))),introduced(assumption,[])).
% 15.70/15.78 thf(h310,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.78 thf(h311,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (kfst @ X3)) @ X1))))),introduced(assumption,[])).
% 15.70/15.78 thf(h312,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => 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=> ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ 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(![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ 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(~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => 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@ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.78 thf(h313,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X1 = X3)))))),introduced(assumption,[])).
% 15.70/15.78 thf(h314,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.78 thf(h315,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X1 = X3)))))),introduced(assumption,[])).
% 15.70/15.78 thf(h316,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.78 thf(h317,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X1) @ emptyset))))),introduced(assumption,[])).
% 15.70/15.78 thf(h318,assumption,(~(((![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => 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@ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.78 thf(h319,assumption,(![X1:$i]:(![X2:$i]:((X1 = X2) => (((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ emptyset))))),introduced(assumption,[])).
% 15.70/15.78 thf(h320,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ 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(powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.79 thf(h321,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => ((X3 = X4) => (X2 = X4))))))),introduced(assumption,[])).
% 15.70/15.79 thf(h322,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.79 thf(h323,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset)) = ((setadjoin @ X3) @ emptyset)) => (X1 = X2))))),introduced(assumption,[])).
% 15.70/15.79 thf(h324,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in 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X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.79 thf(h325,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((setadjoin @ ((setadjoin @ X1) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X1) @ ((setadjoin @ X2) @ emptyset))) @ emptyset)) = ((setadjoin @ ((setadjoin @ X3) @ emptyset)) @ ((setadjoin @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))) @ emptyset))) => (X2 = X4)))))),introduced(assumption,[])).
% 15.70/15.79 thf(h326,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))) => ((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => 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=> (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.79 thf(h327,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:((((kpair @ X1) @ X2) = ((kpair @ X3) @ X4)) => (X2 = X4)))))),introduced(assumption,[])).
% 15.70/15.79 thf(h328,assumption,(~(((![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.80 thf(h329,assumption,(![X1:$i]:((iskpair @ X1) => (singleton @ ((dsetconstr @ (setunion @ X1)) @ (^[X2:$i]:(X1 = ((kpair @ (kfst @ X1)) @ X2))))))),introduced(assumption,[])).
% 15.70/15.80 thf(h330,assumption,(~(((![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))) => ((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.80 thf(h331,assumption,(![X1:$i]:(![X2:$i]:((ksnd @ ((kpair @ X1) @ X2)) = X2))),introduced(assumption,[])).
% 15.70/15.80 thf(h332,assumption,(~(((![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ 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((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.80 thf(h333,assumption,(![X1:$i]:((iskpair @ X1) => (((kpair @ (kfst @ X1)) @ (ksnd @ X1)) = X1))),introduced(assumption,[])).
% 15.70/15.80 thf(h334,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.80 thf(h335,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => ((in @ (ksnd @ X3)) @ X2))))),introduced(assumption,[])).
% 15.70/15.80 thf(h336,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.80 thf(h337,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X3) @ X1)))))),introduced(assumption,[])).
% 15.70/15.80 thf(h338,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => 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((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h339,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(![X4:$i]:(((in @ ((kpair @ X3) @ X4)) @ ((cartprod @ X1) @ X2)) => ((in @ X4) @ X2)))))),introduced(assumption,[])).
% 15.70/15.81 thf(h340,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ 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=> (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h341,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => (((kpair @ X3) @ X4) = ((kpair @ X3) @ X4)))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h342,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h343,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((kfst @ ((kpair @ X3) @ X4)) = X3))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h344,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ 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@ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h345,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X2) => ((ksnd @ ((kpair @ X3) @ X4)) = X4))))))),introduced(assumption,[])).
% 15.70/15.81 thf(h346,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.82 thf(h347,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((cartprod @ X1) @ X2)) => (((kpair @ (kfst @ X3)) @ (ksnd @ X3)) = X3))))),introduced(assumption,[])).
% 15.78/15.82 thf(h348,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => 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X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.82 thf(h349,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((X3 @ X4) @ X5) => ((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)))))))))),introduced(assumption,[])).
% 15.78/15.82 thf(h350,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ 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(powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.82 thf(h351,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((subset @ (((dpsetconstr @ X1) @ X2) @ X3)) @ ((cartprod @ X1) @ X2))))),introduced(assumption,[])).
% 15.78/15.82 thf(h352,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ 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((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ 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(((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.82 thf(h353,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(((breln @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ X3))))),introduced(assumption,[])).
% 15.78/15.82 thf(h354,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ 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(![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in 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X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h355,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h356,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ 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X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ 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@ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h357,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X4) @ X1))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h358,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 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X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h359,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((in @ X5) @ X2))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h360,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h361,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:(![X4:$i]:(![X5:$i]:(((in @ ((kpair @ X4) @ X5)) @ (((dpsetconstr @ X1) @ X2) @ X3)) => ((X3 @ X4) @ X5))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h362,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h363,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3)))))))))),introduced(assumption,[])).
% 15.78/15.83 thf(h364,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = 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X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h365,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ (setunion @ ((dsetconstr @ X2) @ (^[X5:$i]:((in @ ((kpair @ X4) @ X5)) @ X3))))) @ X2))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h366,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h367,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h368,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ 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=> (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ 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X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h369,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (((func @ X1) @ X2) @ X3))))),introduced(assumption,[])).
% 15.78/15.84 thf(h370,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ 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((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ 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=> (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h371,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((ap @ X1) @ X2) @ X3) @ X4)) @ X2))))))),introduced(assumption,[])).
% 15.78/15.84 thf(h372,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h373,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))),introduced(assumption,[])).
% 15.78/15.85 thf(h374,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h375,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((dpsetconstr @ X1) @ X2) @ (^[X4:$i]:((=) @ (X3 @ X4))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h376,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h377,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (((func @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)))))),introduced(assumption,[])).
% 15.78/15.85 thf(h378,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h379,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((in @ (((lam @ X1) @ X2) @ X3)) @ ((funcSet @ X1) @ X2)))))),introduced(assumption,[])).
% 15.78/15.85 thf(h380,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h381,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h382,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i>$o]:((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => (X4 @ ((kpair @ X5) @ X6))))))) => (![X5:$i]:(((in @ X5) @ X3) => (X4 @ X5))))))))) => ((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h383,assumption,(~(((![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h384,assumption,(![X1:$i]:(![X2:$i>$o]:(((ex1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((X2 @ X3) => ((X2 @ X4) => (X3 = X4)))))))))),introduced(assumption,[])).
% 15.78/15.85 thf(h385,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h386,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h387,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h388,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((kpair @ X4) @ ((((ap @ X1) @ X2) @ X3) @ X4))) @ X3))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h389,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h390,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h391,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h392,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3))))))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h393,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h394,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X2) => (((in @ ((kpair @ X4) @ X5)) @ X3) => (((((ap @ X1) @ X2) @ X3) @ X4) = X5)))))))))),introduced(assumption,[])).
% 15.78/15.86 thf(h395,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h396,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((subset @ X3) @ X4)))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h397,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h398,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((breln @ X1) @ X2) @ X3) => (![X4:$i]:((((breln @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X3) => ((in @ ((kpair @ X5) @ X6)) @ X4)))))) => ((![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X2) => (((in @ ((kpair @ X5) @ X6)) @ X4) => ((in @ ((kpair @ X5) @ X6)) @ X3)))))) => (X3 = X4))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h399,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ 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X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ 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(((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h400,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:((((func @ X1) @ X2) @ X4) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h401,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ 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@ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h402,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X1) @ X2)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X5) = ((((ap @ X1) @ X2) @ X4) @ X5)))) => (X3 = X4)))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h403,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ 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((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h404,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h405,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h406,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ X3) @ X4) = ((((ap @ X1) @ X2) @ X3) @ X4)))))))),introduced(assumption,[])).
% 15.78/15.87 thf(h407,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.88 thf(h408,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))),introduced(assumption,[])).
% 15.78/15.88 thf(h409,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ 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((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.88 thf(h410,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:((((func @ X1) @ X2) @ X3) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))),introduced(assumption,[])).
% 15.78/15.88 thf(h411,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.88 thf(h412,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => ((((lam @ X1) @ X2) @ X3) = (((lam @ X1) @ X2) @ X3)))))),introduced(assumption,[])).
% 15.78/15.88 thf(h413,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ X1) => (((((ap @ X1) @ X2) @ (((lam @ X1) @ X2) @ X3)) @ X4) = (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.88 thf(h414,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.88 thf(h415,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((lam @ X1) @ X2) @ (((ap @ X1) @ X2) @ X3)) = X3))))),introduced(assumption,[])).
% 15.78/15.88 thf(h416,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ 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((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ 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X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.89 thf(h417,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => ((in @ X4) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))),introduced(assumption,[])).
% 15.78/15.89 thf(h418,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ 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((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.89 thf(h419,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X4) @ emptyset))))))))),introduced(assumption,[])).
% 15.78/15.89 thf(h420,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.78/15.89 thf(h421,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => ((in @ X3) @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))))))))))),introduced(assumption,[])).
% 15.78/15.89 thf(h422,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h423,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4))))))))) = ((setadjoin @ X3) @ emptyset))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h424,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h425,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (singleton @ ((dsetconstr @ X1) @ (^[X5:$i]:((X2 => (~((X5 = X3)))) => (~(((~(X2)) => (~((X5 = X4)))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h426,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))) => ((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ 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X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h427,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ X1))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h428,assumption,(~(((![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => 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@ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ 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@ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h429,assumption,(![X1:$i]:((singleton @ X1) => (![X2:$i]:(((in @ X2) @ X1) => ((setunion @ X1) = X2))))),introduced(assumption,[])).
% 15.86/15.89 thf(h430,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h431,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (X2 => (((((if @ X1) @ X2) @ X3) @ X4) = X3)))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h432,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))) => ((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h433,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((~(X2)) => (((((if @ X1) @ X2) @ X3) @ X4) = X4)))))))),introduced(assumption,[])).
% 15.86/15.89 thf(h434,assumption,(~(((![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h435,assumption,(![X1:$i]:(![X2:$o]:(![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => ((in @ ((((if @ X1) @ X2) @ X3) @ X4)) @ ((setadjoin @ X3) @ ((setadjoin @ X4) @ emptyset))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h436,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h437,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ X1))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h438,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h439,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((binunion @ X2) @ X3)) @ (powerset @ X1))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h440,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h441,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ (powerset @ X2)) @ (powerset @ (powerset @ X1)))))),introduced(assumption,[])).
% 15.86/15.90 thf(h442,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ 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X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h443,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X2) @ X3)) @ (powerset @ X1))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h444,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.90 thf(h445,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ X1))))),introduced(assumption,[])).
% 15.86/15.90 thf(h446,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h447,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ X2)))) => (X2 = X3)))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h448,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h449,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((subset @ X2) @ X3))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h450,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => 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(powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ 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X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ 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(^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h451,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))) => ((in @ X2) @ (powerset @ X3)))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h452,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h453,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X2) @ (powerset @ X3)) => (((in @ X4) @ X2) => ((in @ X4) @ X3)))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h454,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ 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X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h455,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => (~(((in @ X3) @ ((setminus @ X1) @ X2)))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h456,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h457,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => ((~(((in @ X3) @ ((setminus @ X1) @ X2)))) => ((in @ X3) @ X2))))))),introduced(assumption,[])).
% 15.86/15.91 thf(h458,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h459,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h460,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h461,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h462,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h463,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (((in @ X4) @ X3) => (~(((in @ X4) @ X2)))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h464,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => 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X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h465,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((subset @ X2) @ X3) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ X2)))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h466,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h467,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ X3) => ((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h468,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h469,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ ((setminus @ X1) @ X3)) @ ((setminus @ X1) @ X2)) => ((subset @ X2) @ X3))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h470,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h471,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ X2) => ((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h472,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h473,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ X1) => (((in @ X3) @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) => ((in @ X3) @ X2))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h474,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h475,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ X2) @ ((setminus @ X1) @ ((setminus @ X1) @ X2)))))),introduced(assumption,[])).
% 15.86/15.92 thf(h476,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.92 thf(h477,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ ((setminus @ X1) @ X2))) @ X2)))),introduced(assumption,[])).
% 15.86/15.92 thf(h478,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h479,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (X2 = ((setminus @ X1) @ ((setminus @ X1) @ X2)))))),introduced(assumption,[])).
% 15.86/15.93 thf(h480,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h481,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (~(((in @ X4) @ ((binintersect @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h482,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h483,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h484,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h485,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((subset @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3)))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h486,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h487,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h488,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h489,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ X3) => ((in @ X4) @ ((setminus @ X1) @ X2))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h490,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h491,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h492,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h493,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X2))))))))))),introduced(assumption,[])).
% 15.86/15.93 thf(h494,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h495,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ ((binunion @ X2) @ X3)))) => (~(((in @ X4) @ X3))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h496,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h497,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X2) @ X4)))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h498,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h499,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binunion @ X3) @ X4)))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h500,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h501,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binintersect @ X2) @ X3)) => ((in @ X5) @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h502,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h503,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h504,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h505,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => (~(((in @ X4) @ X2))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h506,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h507,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h508,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h509,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$o]:(![X5:$i]:(((in @ X5) @ X1) => (((in @ X5) @ ((binunion @ X2) @ X3)) => ((((in @ X5) @ X2) => X4) => ((((in @ X5) @ X3) => X4) => X4))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h510,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h511,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => ((~(((in @ X4) @ X2))) => ((~(((in @ X4) @ X3))) => (~(((in @ X4) @ ((binunion @ X2) @ X3))))))))))))),introduced(assumption,[])).
% 15.86/15.94 thf(h512,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h513,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h514,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h515,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h516,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ X3)))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h517,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h518,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))) => ((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h519,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h520,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h521,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h522,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ X2) @ X3)) = ((binunion @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h523,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h524,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) => ((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h525,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h526,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((setminus @ X1) @ X2)) => (((in @ X4) @ ((setminus @ X1) @ X3)) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h527,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h528,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ X4) @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) => ((in @ X4) @ ((setminus @ X1) @ ((binunion @ X2) @ X3))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h529,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h530,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((setminus @ X1) @ ((binunion @ X2) @ X3)) = ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3)))))))),introduced(assumption,[])).
% 15.86/15.95 thf(h531,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h532,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ ((binintersect @ X2) @ X3)) => ((in @ X4) @ X1)))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h533,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h534,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h535,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h536,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X3) @ X4) => ((subset @ ((binintersect @ X2) @ X3)) @ X4))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h537,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h538,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (((subset @ X2) @ X3) => (((subset @ X2) @ X4) => ((subset @ X2) @ ((binintersect @ X3) @ X4))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h539,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h540,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((subset @ X2) @ X4) => (((subset @ X3) @ X5) => ((subset @ ((binintersect @ X2) @ X3)) @ ((binintersect @ X4) @ X5))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h541,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ X2)) @ (powerset @ ((setminus @ X1) @ ((binintersect @ X2) @ X3))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h542,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h543,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => (![X5:$i]:(((in @ X5) @ (powerset @ X1)) => (((setminus @ X1) @ ((binintersect @ ((binunion @ X2) @ X3)) @ ((binunion @ X4) @ X5))) = ((binunion @ ((binintersect @ ((setminus @ X1) @ X2)) @ ((setminus @ X1) @ X3))) @ ((binintersect @ ((setminus @ X1) @ X4)) @ ((setminus @ X1) @ X5))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h544,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (![X4:$i]:(((in @ X4) @ (powerset @ X1)) => ((in @ ((binintersect @ X2) @ X3)) @ (powerset @ ((binintersect @ ((binunion @ X2) @ X4)) @ ((binunion @ X3) @ X4))))))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h545,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => (((subset @ X2) @ ((setminus @ X1) @ X3)) => ((subset @ X3) @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h546,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (powerset @ X1)) => (![X3:$i]:(((in @ X3) @ (powerset @ X1)) => ((in @ ((setminus @ X1) @ ((binunion @ X2) @ X3))) @ (powerset @ ((setminus @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h547,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h548,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h549,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h550,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ (breln1Set @ X1)) => ((breln1 @ X1) @ X2)))),introduced(assumption,[])).
% 15.86/15.96 thf(h551,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h552,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (~((![X5:$i]:(((in @ X5) @ X2) => (~(((X3 @ X4) @ X5))))))))) => (![X4:$i]:(((in @ X4) @ (breln1Set @ X2)) => (((reflwellordering @ X2) @ X4) => (![X5:$i]:(((in @ X5) @ X1) => (singleton @ ((dsetconstr @ X2) @ (^[X6:$i]:(~((((X3 @ X5) @ X6) => (~((![X7:$i]:(((in @ X7) @ X2) => (((X3 @ X5) @ X7) => ((in @ ((kpair @ X6) @ X7)) @ X4))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.96 thf(h553,assumption,(~(((![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h554,assumption,(![X1:$i]:(![X2:$i>$i>$o]:((breln1 @ X1) @ (((dpsetconstr @ X1) @ X1) @ X2)))),introduced(assumption,[])).
% 15.86/15.97 thf(h555,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i>$o]:((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => (X3 @ ((kpair @ X4) @ X5))))))) => (![X4:$i]:(((in @ X4) @ X2) => (X3 @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h556,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h557,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((subset @ X2) @ X3))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h558,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h559,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))) => ((![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ X2)))))) => (X2 = X3)))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h560,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.86/15.97 thf(h561,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => ((breln1 @ X1) @ ((breln1invset @ X1) @ X2))))),introduced(assumption,[])).
% 15.86/15.97 thf(h562,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h563,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X3) @ X4)) @ X2) => ((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h564,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h565,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((in @ X3) @ X1) => (![X4:$i]:(((in @ X4) @ X1) => (((in @ ((kpair @ X4) @ X3)) @ ((breln1invset @ X1) @ X2)) => ((in @ ((kpair @ X3) @ X4)) @ X2))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h566,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h567,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ (((breln1compset @ X1) @ X2) @ X3))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h568,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h569,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (![X6:$i]:(((in @ X6) @ X1) => (((in @ ((kpair @ X4) @ X6)) @ X2) => (((in @ ((kpair @ X6) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h570,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h571,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (~((![X6:$i]:((~((((in @ X6) @ X1) => (~(((in @ ((kpair @ X4) @ X6)) @ X2)))))) => (~(((in @ ((kpair @ X6) @ X5)) @ X3))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h572,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h573,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ (((breln1compset @ X1) @ X2) @ X3)) => (![X6:$o]:((![X7:$i]:(((in @ X7) @ X1) => (((in @ ((kpair @ X4) @ X7)) @ X2) => (((in @ ((kpair @ X7) @ X5)) @ X3) => X6)))) => X6)))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h574,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h575,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => ((breln1 @ X1) @ ((binunion @ X2) @ X3))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h576,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h577,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X2) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h578,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h579,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ X3) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h580,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h581,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)) => ((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h582,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h583,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => ((~(((in @ ((kpair @ X4) @ X5)) @ X2))) => ((in @ ((kpair @ X4) @ X5)) @ X3)))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h584,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h585,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((in @ X4) @ X1) => (![X5:$i]:(((in @ X5) @ X1) => (((in @ ((kpair @ X4) @ X5)) @ ((binunion @ X2) @ X3)) => (![X6:$o]:((((in @ ((kpair @ X4) @ X5)) @ X2) => X6) => ((((in @ ((kpair @ X4) @ X5)) @ X3) => X6) => X6))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h586,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h587,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((binunion @ X2) @ X3) = ((binunion @ X3) @ X2))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h588,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h589,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (X2 = ((breln1invset @ X1) @ ((breln1invset @ X1) @ X2)))))),introduced(assumption,[])).
% 15.94/15.97 thf(h590,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h591,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (((breln1invset @ X1) @ (((breln1compset @ X1) @ X2) @ X3)) = (((breln1compset @ X1) @ ((breln1invset @ X1) @ X3)) @ ((breln1invset @ X1) @ X2)))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h592,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))) => ((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h593,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ (((breln1compset @ X1) @ X2) @ X4)) @ (((breln1compset @ X1) @ X3) @ X4)))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h594,assumption,(~(((![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h595,assumption,(![X1:$i]:(![X2:$i]:(((breln1 @ X1) @ X2) => (![X3:$i]:(((breln1 @ X1) @ X3) => (![X4:$i]:(((breln1 @ X1) @ X4) => ((((breln1compset @ X1) @ ((binunion @ X2) @ X3)) @ X4) = ((binunion @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X3)))) @ ((breln1invset @ X1) @ (((breln1compset @ X1) @ ((breln1invset @ X1) @ X4)) @ ((breln1invset @ X1) @ X2)))))))))))),introduced(assumption,[])).
% 15.94/15.97 thf(h596,assumption,(~(((![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h597,assumption,(![X1:$i]:(![X2:$i>$i]:(~((![X3:$i]:(~((![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h598,assumption,(~(((![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h599,assumption,(![X1:$i]:(![X2:$i>$i]:(exu @ (^[X3:$i]:(![X4:$i]:(((in @ X4) @ X3) = (~((![X5:$i]:(((in @ X5) @ X1) => (~((X4 = (X2 @ X5)))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h600,assumption,(~(((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h601,assumption,(![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) = (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h602,assumption,(~(((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))) => ((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h603,assumption,(![X1:$i]:(![X2:$i>$i]:(![X3:$i]:(((in @ X3) @ ((image1 @ X1) @ X2)) => (~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h604,assumption,(~(((![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h605,assumption,(![X1:$i]:(![X2:$i>$i]:(![X3:$i]:((~((![X4:$i]:(((in @ X4) @ X1) => (~((X3 = (X2 @ X4)))))))) => ((in @ X3) @ ((image1 @ X1) @ X2)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h606,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h607,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((funcSet @ X1) @ X2)) => ((((injective @ X1) @ X2) @ X3) => ((in @ X3) @ ((injFuncSet @ X1) @ X2))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h608,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h609,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h610,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h611,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((injFuncSet @ X1) @ X2)) => (((injective @ X1) @ X2) @ X3))))),introduced(assumption,[])).
% 15.95/15.99 thf(h612,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h613,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => ((in @ X3) @ ((funcSet @ X1) @ X2)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h614,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))) => ((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h615,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i]:(((in @ X3) @ ((surjFuncSet @ X1) @ X2)) => (((surjective @ X1) @ X2) @ X3))))),introduced(assumption,[])).
% 15.95/15.99 thf(h616,assumption,(~(((![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))) => ((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h617,assumption,(![X1:$i]:(![X2:$i]:(![X3:$i>$i]:((![X4:$i]:(((in @ X4) @ X1) => ((in @ (X3 @ X4)) @ X2))) => (![X4:$i]:(((in @ X4) @ ((funcSet @ X2) @ X1)) => ((![X5:$i]:(((in @ X5) @ X1) => (((((ap @ X2) @ X1) @ X4) @ (X3 @ X5)) = X5))) => (((surjective @ X2) @ X1) @ X4)))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h618,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))) => ((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h619,assumption,(![X1:$i]:(![X2:$i]:(((in @ X2) @ ((funcSet @ X1) @ (powerset @ X1))) => (~((((surjective @ X1) @ (powerset @ X1)) @ X2)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h620,assumption,(~(((![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))) => ((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h621,assumption,(![X1:$i]:((nonempty @ X1) => (~((![X2:$i]:(((in @ X2) @ X1) => (~((((binintersect @ X2) @ X1) = emptyset))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h622,assumption,(~(((![X1:$i]:(~(((in @ X1) @ X1)))) => ((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h623,assumption,(![X1:$i]:(~(((in @ X1) @ X1)))),introduced(assumption,[])).
% 15.95/15.99 thf(h624,assumption,(~(((![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h625,assumption,(![X1:$i]:(![X2:$i]:(((in @ X1) @ X2) => (~(((in @ X2) @ X1)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h626,assumption,(~(((![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h627,assumption,(![X1:$i]:(((in @ X1) @ omega) => ((in @ (omegaS @ X1)) @ omega))),introduced(assumption,[])).
% 15.95/15.99 thf(h628,assumption,(~(((![X1:$i]:(((in @ X1) @ omega) => ((in @ ((setadjoin @ X1) @ X1)) @ omega))) => ((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h629,assumption,(~(((![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))) => ((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h630,assumption,(![X1:$i]:(((in @ X1) @ omega) => (~(((omegaS @ X1) = emptyset))))),introduced(assumption,[])).
% 15.95/15.99 thf(h631,assumption,(~(((![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))) => ((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h632,assumption,(![X1:$i]:(((in @ X1) @ omega) => (![X2:$i]:(((in @ X2) @ omega) => (((omegaS @ X1) = (omegaS @ X2)) => (X1 = X2)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h633,assumption,(~(((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2))))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h634,assumption,(![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))))),introduced(assumption,[])).
% 15.95/15.99 thf(h635,assumption,(~((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:((![X3:$i]:(((in @ X3) @ X2) => ((subset @ X3) @ X2))) => (![X3:$i]:(((in @ X3) @ ((binintersect @ X1) @ X2)) => ((subset @ X3) @ ((binintersect @ X1) @ X2)))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h636,assumption,(~(((![X1:$i]:(((in @ X1) @ eigen__0) => ((subset @ X1) @ eigen__0))) => (![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ ((binintersect @ eigen__0) @ X1)) => ((subset @ X2) @ ((binintersect @ eigen__0) @ X1))))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h637,assumption,(![X1:$i]:(((in @ X1) @ eigen__0) => ((subset @ X1) @ eigen__0))),introduced(assumption,[])).
% 15.95/15.99 thf(h638,assumption,(~((![X1:$i]:((![X2:$i]:(((in @ X2) @ X1) => ((subset @ X2) @ X1))) => (![X2:$i]:(((in @ X2) @ ((binintersect @ eigen__0) @ X1)) => ((subset @ X2) @ ((binintersect @ eigen__0) @ X1)))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h639,assumption,(~(((![X1:$i]:(((in @ X1) @ eigen__1) => ((subset @ X1) @ eigen__1))) => (![X1:$i]:(((in @ X1) @ ((binintersect @ eigen__0) @ eigen__1)) => ((subset @ X1) @ ((binintersect @ eigen__0) @ eigen__1))))))),introduced(assumption,[])).
% 15.95/15.99 thf(h640,assumption,(![X1:$i]:(((in @ X1) @ eigen__1) => ((subset @ X1) @ eigen__1))),introduced(assumption,[])).
% 15.95/15.99 thf(h641,assumption,(~((![X1:$i]:(((in @ X1) @ ((binintersect @ eigen__0) @ eigen__1)) => ((subset @ X1) @ ((binintersect @ eigen__0) @ eigen__1)))))),introduced(assumption,[])).
% 15.95/15.99 thf(h642,assumption,(~((((in @ eigen__2) @ ((binintersect @ eigen__0) @ eigen__1)) => ((subset @ eigen__2) @ ((binintersect @ eigen__0) @ eigen__1))))),introduced(assumption,[])).
% 15.95/15.99 thf(h643,assumption,((in @ eigen__2) @ ((binintersect @ eigen__0) @ eigen__1)),introduced(assumption,[])).
% 15.95/15.99 thf(h644,assumption,(~(((subset @ eigen__2) @ ((binintersect @ eigen__0) @ eigen__1)))),introduced(assumption,[])).
% 15.95/15.99 thf(nax1, axiom, (p1<=fsubset @ f__2 @ (fbinintersect @ f__0 @ f__1)), file('<stdin>', nax1)).
% 15.95/15.99 thf(ax228, axiom, ~(p1), file('<stdin>', ax228)).
% 15.95/15.99 thf(pax219, axiom, (p219=>![X1:$i, X2:$i, X3:$i]:(fsubset @ X3 @ X1=>(fsubset @ X3 @ X2=>fsubset @ X3 @ (fbinintersect @ X1 @ X2)))), file('<stdin>', pax219)).
% 15.95/15.99 thf(pax218, axiom, (p218=>![X1:$i, X2:$i, X3:$i]:(fin @ X3 @ (fbinintersect @ X1 @ X2)=>fin @ X3 @ X1)), file('<stdin>', pax218)).
% 15.95/15.99 thf(pax2, axiom, (p2=>fin @ f__2 @ (fbinintersect @ f__0 @ f__1)), file('<stdin>', pax2)).
% 15.95/15.99 thf(ax10, axiom, p219, file('<stdin>', ax10)).
% 15.95/15.99 thf(pax4, axiom, (p4=>![X1:$i]:(fin @ X1 @ f__0=>fsubset @ X1 @ f__0)), file('<stdin>', pax4)).
% 15.95/15.99 thf(ax11, axiom, p218, file('<stdin>', ax11)).
% 15.95/15.99 thf(ax227, axiom, p2, file('<stdin>', ax227)).
% 15.95/15.99 thf(pax214, axiom, (p214=>![X1:$i, X2:$i, X3:$i]:(fin @ X3 @ (fbinintersect @ X1 @ X2)=>fin @ X3 @ X2)), file('<stdin>', pax214)).
% 15.95/15.99 thf(ax225, axiom, p4, file('<stdin>', ax225)).
% 15.95/15.99 thf(pax3, axiom, (p3=>![X1:$i]:(fin @ X1 @ f__1=>fsubset @ X1 @ f__1)), file('<stdin>', pax3)).
% 15.95/15.99 thf(ax15, axiom, p214, file('<stdin>', ax15)).
% 15.95/15.99 thf(ax226, axiom, p3, file('<